Problem 18 - A Graphical Approach to Precalculus with Limits

Question

March the logarithm in Column I with its value in Column II. Remember that \(\log _{a} x\) is the exponent to which a must be raised in onder to obtain \(x\). \(\boldsymbol{I}\) (a) \(\log _{3} 81\) (b) \(\log _{3} \frac{1}{3}\) (c) \(\log _{10} 0.01\) (d) \(\log _{6} \sqrt{6}\) (e) \(\log _{e} 1\) (f) \(\log _{3} 27^{3 / 2}\) \(\mathbf{II}\) A. \(-2\) B. \(-1\) C. 0 D. \(\frac{1}{2}\) E. \(\frac{9}{2}\) F. 4

Step-by-Step Solution

Answer

(a) F, (b) B, (c) A, (d) D, (e) C, (f) E.

Step 1: Solve \( \log_3 81 \)

To find \( \log_3 81 \), we need to determine the power to which 3 must be raised to get 81. Recognize that \( 81 = 3^4 \), therefore, \( \log_3 81 = 4 \). This corresponds to option F.

Step 2: Solve \( \log_3 \frac{1}{3} \)

For \( \log_3 \frac{1}{3} \), we are looking for the power to which 3 must be raised to get \( \frac{1}{3} \). Notice that \( \frac{1}{3} = 3^{-1} \), so \( \log_3 \frac{1}{3} = -1 \). This corresponds to option B.

Step 3: Solve \( \log_{10} 0.01 \)

For \( \log_{10} 0.01 \), we find the power to which 10 must be raised to result in 0.01. Since \( 0.01 = 10^{-2} \), it follows that \( \log_{10} 0.01 = -2 \). This matches option A.

Step 4: Solve \( \log_6 \sqrt{6} \)

For \( \log_6 \sqrt{6} \), recall that \( \sqrt{6} = 6^{1/2} \). Thus, \( \log_6 \sqrt{6} = \frac{1}{2} \). This is option D.

Step 5: Solve \( \log_e 1 \)

When solving \( \log_e 1 \), note that any logarithm with a base raised to the power of 0 equals 1. Therefore, \( \log_e 1 = 0 \). This corresponds to option C.

Step 6: Solve \( \log_3 27^{3/2} \)

In the case of \( \log_3 27^{3/2} \), start by rewriting 27 as a power of 3. Since 27 is \( 3^3 \), the expression becomes \( (3^3)^{3/2} = 3^{9/2} \). Thus, \( \log_3 27^{3/2} = \frac{9}{2} \). This is option E.

Key Concepts

Exponents

Exponents are a mathematical concept that represent repeated multiplication of a number by itself. For example, when we say - \( 3^4 \), it means that 3 is multiplied by itself 4 times: \( 3 \times 3 \times 3 \times 3 = 81 \). Exponents are crucial in understanding logarithms because they allow us to express large or small numbers in a compact form. They show up frequently in science, engineering, and everyday calculations. By grasping exponents, you can solve problems involving exponential growth, compound interest, and scientific calculations. Exponents also follow certain rules, such as:

• Product of powers: \( a^m \times a^n = a^{m+n} \)
• Quotient of powers: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a power: \( (a^m)^n = a^{m \times n} \)

Understanding these basics helps you manipulate and solve equations more efficiently.

Base

The base in mathematics, especially in logarithms and exponents, is the number that is raised to a power. In the expression \( b^n \), \( b \) is the base and \( n \) is the exponent. The base tells you what number is being multiplied by itself. In logarithms, the base is pivotal as it signifies the number that needs to be raised to a certain power to obtain another number. For instance, in \( \log_3 81 \), 3 is the base.Different bases are applicable in varying contexts:

• Base 10 (common logarithm): Often used in scientific calculators and is represented as \( \log \).
• Base \( e \) (natural logarithm): Used in calculus and exponential growth, denoted as \( \ln \).
• Base 2: Relevant in computer science and binary systems.

Understanding the base is essential when working with logarithms because it determines how the logarithmic relationship is structured and used.

Logarithmic Equations

Logarithmic equations involve expressions that contain logarithms. They often require us to find the unknown value by converting a logarithmic equation to its exponential form.To solve a logarithmic equation like \( \log_b x = y \), we can rewrite it in exponential form: \( b^y = x \). This allows us to easily solve for the unknown variable, make calculations, or simplify expressions.Here are steps to tackle logarithmic equations:

• Identify the base of the logarithm.
• Rearrange the equation in exponential form if needed.
• Solve the resulting equation for the unknown.

Logarithmic equations have vast applications in fields like finance (to solve for time in compound interest), acoustics (to measure sound levels), and earthquake intensity (Richter scale). Mastering logarithmic equations enables you to connect theoretical understanding with practical real-world problems.

Properties of Logarithms

The properties of logarithms are tools that simplify complex logarithmic expressions, making it easier to solve equations and interpret data.Here are some key properties:

• Product rule: \( \log_b (MN) = \log_b M + \log_b N \)
• Quotient rule: \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \)
• Power rule: \( \log_b (M^n) = n \cdot \log_b M \)
• Change of base formula: \( \log_b M = \frac{\log_k M}{\log_k b} \)

These properties allow for the flexibility needed when manipulating logarithms. For example, the power rule is used frequently in calculus to derive expressions and simplify differentiation. The change of base formula is particularly helpful when the base of a logarithm isn't readily calculator-compliant.By understanding and applying these properties, solving logarithmic equations becomes much more straightforward, enabling efficient and precise problem-solving.